Special Right Triangles

5.1 (M2)

Pythagorean Theorem

Right Triangle Theorems

45o-45o-90o Triangle Theorem Hypotenuse is times as long as each

leg

30o-60o-90o Triangle Theorem Hypotenuse is twice as long as the

shorter leg, and the longer leg is times as long as the shorter leg

2

3

EXAMPLE 1 Find hypotenuse length in a 45-45-90 triangleo o o

Find the length of the hypotenuse.

a.

SOLUTION

hypotenuse = leg 2

= 8 2 Substitute.

45-45-90 Triangle Theoremo o o

By the Triangle Sum Theorem, the measure of the third angle must be 45º. Then the triangle is a 45º-45º-90º triangle, so by Theorem 7.8, the hypotenuse is 2 times as long as each leg.

a.

EXAMPLE 1 Find hypotenuse length in a 45-45-90 triangleo o o

hypotenuse = leg 2

Substitute.

45-45-90 Triangle Theoremo o o

= 3 2 2= 3 2 Product of square roots

= 6 Simplify.

b. By the Base Angles Theorem and the Corollary to the Triangle Sum Theorem, the triangle is a triangle.

45 - 45 - 90ooo

Find the length of the hypotenuse.b.

EXAMPLE 2 Find leg lengths in a 45-45-90 triangleo o o

Find the lengths of the legs in the triangle.

SOLUTION

By the Base Angles Theorem and the Corollary to the Triangle Sum Theorem, the triangle is a triangle.

45 - 45 - 90ooo

hypotenuse = leg 2

Substitute.

45-45-90 Triangle Theoremo o o

25 = x 2

252

=2x2

5 = x

Divide each side by 2

Simplify.

EXAMPLE 3 Standardized Test Practice

SOLUTION

By the Corollary to the Triangle Sum Theorem, the triangle is a triangle.45 - 45 - 90

ooo

EXAMPLE 3 Standardized Test Practice

hypotenuse = leg 2

Substitute.

45-45-90 Triangle Theoremo o o

= 25 2WX

The correct answer is B.

GUIDED PRACTICE for Examples 1, 2, and 3

Find the value of the variable.

1. 2. 3.

ANSWER 2 ANSWER 2 8 2ANSWER

GUIDED PRACTICE for Examples 1, 2, and 3

4. Find the leg length of a 45°- 45°- 90° triangle with a hypotenuse length of 6.

3 2ANSWER

EXAMPLE 4 Find the height of an equilateral triangle

Logo

The logo on a recycling bin resembles an equilateral triangle with side lengths of 6 centimeters. What is the approximate height of the logo?

SOLUTION

Draw the equilateral triangle described. Its altitude forms the longer leg of two 30°-60°-60° triangles. The length h of the altitude is approximately the height of the logo.

h = 3 5.2 cm

3

longer leg = shorter leg 3

EXAMPLE 5 Find lengths in a 30-60-90 triangleooo

Find the values of x and y. Write your answer in simplest radical form.

STEP 1 Find the value of x.

longer leg = shorter leg 39 = x 3

93 = x

93

33

= x

93

3 = x

3 3 = x Simplify.

Multiply fractions.

Triangle Theorem30 - 60 - 90ooo

Divide each side by 3Multiply numerator and denominator by 3

Substitute.

EXAMPLE 5 Find lengths in a 30-60-90 triangleooo

hypotenuse = 2 shorter leg

STEP 2 Find the value of y.

y = 2 3 = 63 3 Substitute and simplify.

Triangle Theorem30 - 60 - 90o oo

EXAMPLE 6 Find a height

Dump Truck

The body of a dump truck is raised to empty a load of sand. How high is the 14 foot body from the frame when it is tipped upward at the given angle?

a. 45 angleo

b. 60 angleo

SOLUTIONWhen the body is raised 45 above the frame, the height h is the length of a leg of a triangle. The length of the hypotenuse is 14 feet.

a.45 - 45 - 90

ooo

o

EXAMPLE 6 Find a height

14 = h 2142

= h

9.9 h

Triangle Theorem45 - 45 - 90ooo

Divide each side by 2

Use a calculator to approximate.

When the angle of elevation is 45, the body is about 9 feet 11 inches above the frame.

o

b. When the body is raised 60, the height h is the length of the longer leg of a triangle. The length of the hypotenuse is 14 feet.

30 - 60 - 90ooo

o

EXAMPLE 6 Find a height

hypotenuse = 2 shorter leg Triangle Theorem30 - 60 - 90o oo

14 = 2 s Substitute.

7 = s Divide each side by 2.

longer leg = shorter leg 3 Triangle Theorem30 - 60 - 90ooo

h = 7 3 Substitute.

h 12.1 Use a calculator to approximate.

When the angle of elevation is 60, the body is about 12 feet 1 inch above the frame.

o

GUIDED PRACTICE for Examples 4, 5, and 6

Find the value of the variable.

ANSWER 3 ANSWER 3 2

GUIDED PRACTICE for Examples 4, 5, and 6

What If? In Example 6, what is the height of the body of the dump truck if it is raised 30° above the frame?

7.

ANSWER 7 ft

SAMPLE ANSWER

The shorter side is adjacent to the 60° angle, the longer side is adjacent to the 30° angle.

In a 30°- 60°- 90° triangle, describe the location of the shorter side. Describe the location of the longer side?

8.